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Thursday, March 29, 2012

What to Do with All the Technologies

Among people who know me professionally I have a bit of a reputation as a technology person. It might be because when people walk by my room, there's usually laptops and calculators all over the place. Or it might be because I put it all over my CV because people like that sort of thing. Or it might be my third robotic arm. Who knows.

But when people talk to me about the technology I have to constantly Reframe the Issue and explain how I'm not all pro any technology for its own sake. You don't go, "Oh here's this cool technology let me shoehorn it into my classroom." Instead you go, "I think I have thought of the best way to teach this, and it would be impossible in an analog world, but I know enough about the technologies to realize this idea." You don't go to a twenty-minute inservice about xyz.com and go "I'm going to make an xyz.com lesson." You use xyz.com for your own purposes, or you suspect its utility and put it in your back pocket, until your awesome instruction idea needs xyz.com in order to exist. Your lesson is the fuel and xyz.com is the oxygen.

So here is a lesson that would not exist without dynamic geometry software and classroom polling. It does not matter what sort of dynamic geometry software. I've done it with Sketchpad and I've done it with Nspire and next year I'll probably do it with Geogebra (damn, that sentence makes me sound disreputablllllle.) But I don't think you could get the same effect without the technology. Maybe you could give them diagrams on paper and rulers and protractors, but there's no way to make those not static, even if there's a lot of them.

We are discovering additional properties of special kinds of parallelograms. So everyone starts with a sketch of a parallelogram (that you give them, or in our case, that we constructed the day before.) And the children are in groups. Each group gets a different set of questions to explore. Writing these exploration-y questions is a bit of a dark art. You don't want to send them on a chase of the wild-goose variety but you don't want to set them down too much of a pre-defined path either.

Example: Group 1. Start with your sketch of a parallelogram. Construct both diagonals. Drag points around until the diagonals are perpendicular to each other. You will have to decide what to measure so you can be sure. What is the name of the special kind of parallelogram you get as a result of perpendicular diagonals? Now you need to find at least two NEW properties of this shape. They must be NEW properties that are NOT properties of any old parallelogram. You will have to measure some stuff. If you can't find two new properties, keep staring at it until you get a good idea. Write them down. Verify them with measurements. See if your group members agree. Drag vertices to make a different parallelogram where the diagonals are still perpendicular. Are your new properties still true? Challenge: It is possible to construct a quadrilateral with perpendicular diagonals that is NOT a parallelogram. Open a new page and construct such a shape. What other properties does it have?

They get ten or so minutes to play around. It is helpful to give them some verbal marginally-hysterical (at least in my case, they always feel slightly deranged) instructions like "It is not a square! Nobody has squares! The correct answer to any question is not "square!"" and you also need to run around like a crazy person and make sure everyone knows how to grab points and drag them around (you might as well admit that screaming about rhombuses to a roomful of 15 year olds makes you a little bit of a crazy person.) Because you KNOW there are at least three maybe four kids who try one thing for half a second and it doesn't work and then they will sit there and stare at their desk for twenty minutes unless you interrupt that little party.

Once they have had ample time to explore, and the faster workers are getting bogged down in the challenge questions you put at the end, you ask them to respond. I have queried each group verbally in front of the class, one by one, and that doesn't work so hot. Usually in a class of 30+ hardly anybody likes to talk in front of everybody. Better... send them a link to a Google Form where they can type an answer to each question. Or come up with your own response system that your existing tech will support. This year I used TI Navigator polls. They are annoying because the TI's don't have a qwerty keyboard. (Lesson number 9,125,698 learned the hard way.)

Once all the groups have had a chance to report by whatever method, then you write down your notes of properties of rhombuses and rectangles. And then you give them a bunch of problems to find missing measurements in rhombuses and rectangles. They can reason it out now. You don't have to show them example problems first. It feels like kind of a magic trick.

Good question - I struggle with that daily. But it's almost always possible to write questions that can't be solved just by punching something in a calculator. In your function transformations example, they can only "just graph them on the calculator" if they are given an equation. So, you can give them the graph, or a verbal description of a transformation, and ask them to come up with an equation. (Although they could use the calculator to check if their equation did the job - I'm pretty ok with that though.) Or you could give them a piecewise graph of an f(x) which is not any recognizable basic function, and ask them to graph -2f(x+1).

That is a sort of asking-the-same-question-backwards approach. Other ideas: give more open-ended tasks maybe "design a function with these characteristics" that have many possible correct solutions.

Or: ask them to demonstrate or verify something, for example "logically and clearly show how you know it is true that the indefinite integral of xcos(x)dx is xsin(x)+cos(x)+C."

Writing better assessment questions is a big part of what we talked about. Also moving away from answers and towards explanations to gauge student understanding. Then techniques like you're talking about work so well both for assessment and spurring on student activity.

Graham Wegner has mentioned this quote on his blog, so I thought I'd just bring my comment from there over here, seems it refers to the same article!

'Spot on. It’s having that arsenal of tools- digital or analogue- that keep up with the times, that open up possibilities, and that provide inspiration as to how best we can deliver the knowledge we want to get across.